Bulletin T.CXXXIII de l’Académie serbe des sciences et des arts − 2006
Classe des Sciences mathématiques et naturelles
Sciences mathématiques, No 31
ON A MODEL EQUATION THAT REFLECTS SOME OF THE SHEAR FLOW
HYDRODYNAMIC STABILITY PROPERTIES1
V. D. DJORDJEVIĆ
(Presented at the 1st Meeting, held on February 24, 2006)
A b s t r a c t. A model equation is proposed in the paper that mimics
some of the shear flow hydrodynamic stability properties. It contains the basic
velocity profile, which can be arbitrarily chosen, and a nonlinear term, whose
form can be appropriately adjusted to any particular problem. Full linear and
weakly nonlinear theories for the Bickley jet velocity profile are elaborated.
The solution of the linear problem is obtained in terms of associated Legendre
functions. Within the weakly nonlinear theory a Landau equation is derived
that describes the evolution of the perturbations near the critical wave number. The conditions for supercritical stability and subcritical instability are
revealed.
AMS Mathematics Subject Classification (2000): 76E05, 76E30
Key Words: model equation, shear flows, linear stability theory, weakly
nonlinear stability theory, Landau equation
1
This paper was presented at the International Symposium on Nonconservative and
Dissipative Problems in Mechanics, Novi Sad, September 11–14, 2005
30
V. D. Djordjević
1. Introduction
It is well known that two distinct regimes of flow exist in fluid dynamics.
They are laminar and turbulent flows. In laminar flow regime trajectories
of individual fluid particles are smoothly adjusted to the flow boundaries,
and are almost parallel to them. Also, laminar flow can become steady under some conditions. In contrast, turbulent flow is always unsteady, and
the trajectories of fluid particles, in addition to some mean motion, which
is accommodated with the form of fluid boundaries, are accompanied with
intensive irregular fluctuations in the direction perpendicular to the mean
flow. These fluctuations make the turbulent flow chaotic. The differences between laminar and turbulent regimes of flow have far reaching consequences
upon the pressure drop in channels and pipes, the drag of bodies moving
through fluid, heat exchange between the fluid and the boundaries, etc. The
transition from laminar to turbulent flow usually takes place in the same flow
field, and is seldom sudden, in the sense that a clear boundary between the
two regimes can be drawn. Rather, the transition occurs by passing through
several different phases, so that the mathematical description of each of them
is very important for the understanding of turbulence.
Hydrodynamic stability theory deals with the laminar-turbulent transition in various fluid flows. The first step in this theory is usually a linear
theory. Then, it is followed by the so-called weakly nonlinear theory, and
this theory is then followed by the secondary stability theory, etc. It turns
out within this theory that there is no unique route to turbulence. Namely,
a system may pass to turbulence in many different ways depending on the
value of some governing parameters (the feature of inhomogeneity!). Also,
very different systems may experience the transition in exactly the same way
(the feature of universality!). In any case the problems of the hydrodynamic
stability theory are delicate and perplex, and they are algebraically tedious
and involved. Very often the important physics of the problem is hidden
behind the complex mathematical operations. That is why there exist in
the literature several so-called model equations that are not physically related directly to any problem of fluid mechanics, but mimic very accurately
hydrodynamic stability properties of various flows. These properties can be
revealed by much simpler mathematical methods, so that these equations
serve as a very good means for the demonstration of the capabilities of the
theory. Usually they bear the name of their authors, like Eckhaus equation
[1], Segel equation [2], Swift-Hohenberg equation [3], Proudman-Johnson
equation [4], Matkowski equation [5], and others. For example Matkowski
On a model equation that reflects some of the shear flow
31
equation reads
ut − sin u =
1
uxx ,
R
t ≥ 0, 0 ≤ x ≤ π
As a rule, in all of these equations the stability of some trivial solution is
investigated (u = 0, in the case of Matkowski equation), and there is no
possibility to chose another, nontrivial solution. In order to overcome this
shortcoming of all the existing model equations, we propose in this paper an
equation, which contains the basic velocity profile whose stability properties
are investigated, and which can be arbitrarily chosen. It reads:
1 uxx + uyy − u0 ,
R
t ≥ 0, |x| < ∞, a ≤ y ≤ b, u0 = u0 (y), f (0) = 0, R > 0.
ut − u0 f (u − u0 ) =
(1)
As usual, t is time, x and y are Cartesian coordinates, and u is the velocity
in the direction of x. The basic velocity profile in Eq. (1) is denoted by u0 (y)
, and it is obviously a nontrivial solution of the equation, provided f (0) = 0.
It represents a parallel shear flow. At that the constants a and b can be finite
or infinite, i.e., the flow can be unbounded from both sides, bounded from
one side, or bounded from both sides. R is the positive parameter that plays
the role of a Reynolds number, and f is an arbitrary function, supposedly
odd, which in general makes the Eq. (1) nonlinear, and which allows the
following Taylor series representation:
f = f (0) (u − u0 ) +
f (0)
(u − u0 )3 + . . .
3!
(2)
Thus, not only the basic velocity profile can be arbitrarily chosen, but also
the form of the nonlinear term in Eq. (1) can be conveniently adjusted to
the considered problem by the choice of the coefficients f (0), f (0),. . .
It will be shown in the paper that the Eq. (1) is particularly suited for
studying linear and nonlinear stability properties of free and bounded shear
flows. Some of them, which play a very important role in fluid mechanics,
are shown in Fig. 1. They are: mixing layer, jet, wake, boundary layer, and
channel flow. Both, linear and weakly nonlinear stability theories for these
flows are elaborated and presented. It is shown within the linear theory that
the unbounded jet type velocity profile experiences long wave instability,
and that the eigenfunctions are expressed in terms of associated Legendre
functions. Within the weakly nonlinear theory neutral eigen mode is perturbed by introduction of some slowly varying independent variables, and a
32
V. D. Djordjević
Landau type equation is derived, which describes the long time evolution of
this mode. The conditions for the appearance of supercritical stability and
subcritical instability are illuminated.
y
y
y
(a)
(b)
u(y)
(c)
y
y
111111111111111
000000000000000
111111111111111
000000000000000
000000000000000
111111111111111
(d)
u(y)
u(y)
(e)
u(y)
111111111111111
000000000000000
000000000000000
111111111111111
111111111111111
000000000000000
u(y)
111111111111111
000000000000000
111111111111111
000000000000000
111111111111111
000000000000000
Figure 1. Various characteristic shear flow velocity profiles: (a) mixing layer, (b)
jet, (c) wake, (d) boundary layer, and (e) channel or pipe flow
2. Linear theory
Within the linear theory we linearize Eq. (1) by taking the first term in
the expansion (2) only, and present the solution of (1) in the form
u = u0 (y) + û(t, x y),
where û(t, x y) is the small perturbation of the basic velocity profile u0 (y) .
The equation to be satisfied by this perturbation reads
ût − u0 f (0)û =
1
(ûxx + ûyy ) .
R
(3)
33
On a model equation that reflects some of the shear flow
The solution of Eq. (3) that suffices for our purposes is sought in the form
of a single Fourier component (normal mode approach [6])
û = Re A U (y) exp [iα(x − ct)] ,
(4)
in which A is an arbitrary complex constant, U (y) is the complex amplitude
of the wave, α is the real and positive wave number (α = 2π/λ where λ is the
wave length), and c = cr +ici is the complex speed of the wave. Obviously, cr
is the speed of the wave, and α ci is its growth rate. ci > 0 implies instability,
ci < 0 implies stability, while ci = 0 implies neutral stability. Inserting (4)
into Eq. (3) we obtain
U + i α cR − α2 + R f (0)u0 U = 0.
(5)
Together with vanishing boundary conditions on the boundaries: U (a) =
U (b) = 0, Eq. (5) represents the classical Sturm-Liouville eigenvalue problem, which in addition to the eigenfunction offers also an eigenvalue relation
of the form: c = c(α; R). Before we present a concrete solution of this
equation for some of the profiles shown in Fig. 1, we will derive a general
statement concerned with its eigenvalues. Multiplying Eq. (5) with the complex conjugate U ∗ , integrating between a and b, and applying the boundary
conditions, we get the following relation:
b
b
2
2
i α cR − α + R f (0)u0 |U | dy − |U |2 dy = 0,
a
a
from which there immediately follows that cr = 0 . Thus, the perturbations are stationary. Also, the growth rate of the perturbations satisfies the
relation
b
b
b
2
2
2
αci R
|U | dy =
Rf (0)u0 − α |U | dy − |U |2 dy,
a
a
a
and in general it can be both positive and negative. For ci = 0 we may derive
an expression for the critical wave number αc that describes the neutral
eigenmode
b
b
b
αc2
|U |2 dy = R f (0)
u0 |U |2 dy − |U |2 dy.
a
a
a
We will now state a concrete solution for the jet-type flow defined as
u0 = Uo sech2 y, |y| < ∞, (Bickley jet), where U0 is an arbitrary positive constant. For that purpose we will transform Eq. (5) by introducing a
34
V. D. Djordjević
new independent variable: T = tanh y, thus reducing the range of its variations to |T | ≤ 1. By using some of the elementary properties of hyperbolic
functions, for example that: sech2 y ≡ S 2 = 1 − T 2 , Eq. (5) becomes:
d2 U
dU
i α cR − α2
1−T
U = 0,
+ RU0 f (0) +
− 2T
dT 2
dT
1 − T2
2
(6)
with the boundary conditions: U (1) = U (−1) = 0. This is the associated
Legendre equation, whose theory is well established (s. [7], [8]). The only
solutions of this equation that can satisfy the prescribed boundary conditions are those for which R U0 f (0) = N (N + 1), where N is a positive
integer, and for which i α cR − α2 = −μ2 , where μ = n = 1, 2, . . . N . By
this choice of the governing parameters we confine ourselves to the discrete
spectrum of eigenvalues only, and to positive values of the parameter f (0).
The eigenfunctions of the continuous spectrum develop singularities at the
boundaries, and cannot be used. The eigenfunctions corresponding to the
chosen discrete spectrum are [7]:
PNn (T ) = (−1)n (1 − T 2 )n/2
dn PN (T )
= U [T (y)] ,
dT n
dN
1
(T 2 − 1)N are Legendre polynomials. It is obvious
where PN = N
2 N ! dT N
that for all N , cr = 0 , in accordance with the previously derived general
statement. For N = 1 (μ = 1) the growth rate of the perturbations is
determined by α ci R = 1 − α2 , so that the critical wave number is αc = 1,
and the eigenfunction is U = −S. For N = 2 there are two modes. For
μ = 1 the mode is an odd one. The eigenvalue relation and the critical wave
number have the same values as in the previous case, while the eigenfunction
is U = −3ST . For μ = 2 the mode is even. The eigenvalue relation reads
α ci R = 4 − α2 . The critical wave number is αc = 2, and the corresponding
eigenfunction is U = 3S 2 . For N = 3 there are three modes, etc. It depends
on the initial conditions which of the modes will be excited, but we will not
treat that problem here.
A general dependence of the growth rate of the perturbations on the
imposed wave number is sketched in Fig. 2. For α < αc (the dotted part
of the abscissa) the flow is unstable, while for α > αc the flow is stable.
Thus, the jet type profile in the form of a Bickley jet experiences a long
wave instability − exactly as in the classical hydrodynamic stability theory
[6].
35
On a model equation that reflects some of the shear flow
α ci R
αc
α
Figure 2. Stability diagram for a Bickley jet,
as revealed by linear theory
3. Weakly nonlinear theory
Within the weakly nonlinear theory we suppose that the wave number α
differs slightly from its critical value αc , i.e., |αc − α| 1. At that, α can
be both less than αc (linearly unstable case) and greater than αc (linearly
stable case). If α is less than αc but close to it, perturbations will grow
slowly with time. During a long period of time they can become large due to
accumulated effect of nonlinearity, so that the linear theory breaks. If α > αc
perturbations decay with time very slowly, so that again nonlinearity may
come into play and disrupt the basic result of the linear theory. In both cases
a new theory is necessary for the description of long time evolution of such
disturbances, and this is the problem the weakly nonlinear theory is dealt
with.
In order to develop this theory for the jet type velocity profile elaborated
earlier, we will now introduce a small parameter ε defined as the ratio between the maximum amplitude of the perturbations and the thickness of the
velocity profile, and suppose for convenience that |αc − α| = O(ε2 ). In order
to make this assumption explicit we will introduce a slow coordinate ξ = ε2 x
and a slow time τ = ε2 t. This form of the slow variables will be justified
later. Now we will formally seek the solution of Eq. (1) with the nonlinear
term (2) in the form of the following asymptotic series
u = u0 (y) + ε u1 (t, x, τ, ξ, y) + ε2 u2 (t, x, τ, ξ, y) + . . .
By inserting this series into Eq. (1) and equating the terms with the same
power of ε we obtain a system of recursive partial differential equation for
36
V. D. Djordjević
the functions u1 , u2 , . . . The equation for u1 reads:
u1yy + u1xx + R f (0) u0 (y) u1 = 0.
(7)
For the Bickley jet u0 = U0 S 2 , and for the mode N = 1: R f (0) U0 = 2,
and αc = 1. Having in mind the result of the linear theory that for α = αc ,
cr = ci = 0, the solution of Eq. (7) will now have the form:
u1 = Re A(τ, ξ) U1 (y) exp(ix),
(8)
with: U1 = S. The difference between (8) and (4) is that A is not a constant
any more, but a function of slow variables τ and ξ. In what follows we will
derive an equation for the amplitude A(τ, ξ).
At the second order we obtain the following equation for u2 :
u2yy + u2xx + R f (0) u0 u2 = 0
The equation is homogeneous, and there is no interaction between its solution
and the first harmonic- fundamental (8). Thus, its solution is of no interest
for our purposes. At the next order we get
f (0)
3
u0 u1 − 2u1xξ
(9)
u3yy + u3xx + R f (0) u0 u3 = R u1τ −
3!
If evaluated by using (8), the right hand side (r.h.s.) of this equation reads
r.h.s. = Re (R Aτ − 2 i Aξ ) U1 − R f 8(0) |A|2 A u0 U13 exp(ix)
−R f 24(0) u0 Re A3 U13 exp(3ix),
so that the structure of its solution is to be
(1)
(2)
(p)
u3 = Re K1 (τ, ξ) U3 (y) + K2 (τ, ξ) U3 (y) + U3 (τ, ξ, y) exp(ix)
+3rd harmonic term.
(1)
(2)
Here, U3 (y) and U3 (y) represent the two linearly independent solutions of the homogeneous part of Eq. (9), K1 (τ, ξ) and K2 (τ, ξ) are the
(p)
"constants" of integration, and U3 is the particular integral. As indicated earlier, we are primarily interested in the derivation of an equation
for A(τ, ξ), rather than in the evaluation of the small corrections u2 , u3 , . . .
37
On a model equation that reflects some of the shear flow
to u1 . This equation is obtained as a solvability condition for the equation
(p)
for U3 :
f (0)
(p)
|A|2 A S 5 ,
= (R Aτ − 2 i Aξ ) S −
L U3
4 f (0)
where L is a self-adjoint operator
L=
d2
+ (2S 2 − 1).
dy 2
The solvability condition for this equation is obtained as [9]:
∞
∞
(p) ∞ (0)
f
dU
(p) ∞
3
(R Aτ −2 i Aξ ) S 2 dy− |A|2 A S 6 dy = S
+ S T U3
,
4 f (0)
dy
−∞
−∞
−∞
−∞
(p)
(p)
or, after applying the boundary conditions: U3 (∞) = U3 (−∞) = 0, and
evaluating the necessary integrals:
R Aτ − 2 i Aξ −
2f (0)
|A|2 A = 0.
15f (0)
(10)
This is the desired evolution equation. In view of the statement of this
problem, as given at the beginning of this Section, we will use the following
form of its solution
A = |A| exp(−i Δα ξ),
where Δα = αc − α = 1 − α, and |A| = f (τ ). For convenience we will
√
2τ
, |A| = 15 B(τ̃ ) and
introduce a simple transformation of variables: τ̃ =
R
get
f (0) 3
B ,
(11)
Ḃ = Δα B + f (0)
with an arbitrary initial condition: B(0) = B0 > 0.
The Eq. (11) is called the Landau equation in the literature, and its
theory is well established [4]. For Δα = 0 the solution is
B2 = Δα +
f (0)
f (0)
B02
Δα B02
f (0) 2
exp(−2 Δα τ̃ ) − B
f (0) 0
,
(12)
while for Δα = 0 it reads
B2 =
B02
f (0) 2
B τ̃
1−2 f (0) 0
(13)
38
V. D. Djordjević
Qualitative behavior of the solution is shown in Fig. 3, in which arrows
symbolically designate the direction in which the amplitude advances with
time. It is seen that it crucially depends on the sign of f (0).
B
B
* * * * * *
(a)
(b)
** * *
Beq
Δα
Δα
Figure 3. Pitchfork bifurcation for a Bickley jet, as revealed by the weakly
nonlinear theory
For f (0) < 0 (s. Fig. 3a) and Δα > 0 (linearly unstable case) the
amplitude of the perturbations tends to an equilibrium value:
2
=−
Beq
Δα f (0)
.
f (0)
Thus, if initially B < Beq perturbations increase with time, but this increase
is not infinite. If initially B > Beq , then B decreases with time, again tending
to reach the equilibrium (saturated!) value of Beq . Such a flow is called
supercritically stable, and the effect of nonlinearity is obviously stabilizing
in this case. For Δα ≤ 0 (linearly stable case) the perturbations tend to
disappear independently of the initial value.
For f > 0 (s. Fig. 3b) and Δα > 0 there is no stabilizing effect
of nonlinearity - perturbations increase with time for any initial B0 . For
Δα < 0, however, they diminish in accordance with the linear theory if
B02 < −
only, and increase if
B02 > −
Δα f (0)
f (0)
Δα f (0)
.
f (0)
In such a case the flow is said to be subcritically unstable. The "stars" above
the arrows in Fig. 3b indicate that B becomes infinite in finite time, which
On a model equation that reflects some of the shear flow
39
can be readily revealed from the solutions (12) and (13). Such a singularity
of the solution is called the blow-up. Of course, when B attains large values,
weakly nonlinear theory ceases to be applicable, and another, fully nonlinear
theory should be employed. Anyhow, the existence of such a singularity
indicates a relatively fast transition to turbulence. As well known [4] the
qualitative change in the behavior of the solution of a differential equation
when a parameter passes through a characteristic value is called bifurcation.
The bifurcation of the solution of Eq. (1) shown in Fig. 3 is known as the
pitchfork bifurcation.
4. Conclusion
The model equation proposed in this paper is not physically based on
the hydrodynamic stability theory. However, it contains an arbitrary basic
velocity profile, whose stability properties are investigated. Also, some parameters in the nonlinear term of the equation can be conveniently chosen
so as to demonstrate some of the features of the stability properties of shear
flows, for which the equation is particularly suitable. It must be admitted,
however, that not all of these properties are reproducible. For example, the
role of the inflection point on the profile, the existence of the critical layers, the mean flow correction, etc. [6] are missing. Still, we think that the
equation can serve as an effective mean for a relatively simple demonstration of the mathematical treatment of perplex linear and weakly nonlinear
hydrodynamic stability theories.
REFERENCES
[1] W. E c k h a u s, Studies in nonlinear stability theory, Springer Tracts in Natural
Philosophy, Vol. 6, Springer Verlag, Berlin, 1965.
[2] L. A. S e g e l, The structure of nonlinear cellular solutions to the Boussinesq equations,
J. Fluid Mech. 21 (1965), pp. 345–348.
[3] J. S w i f t, P. C. H o h e n b e r g, Hydrodynamic fluctuations at the convective
instability, Phys. Rev. A 15 (1975), pp. 319–328.
[4] P. G. D r a z i n, Nonlinear Systems, Cambridge Texts in Applied Mathematics,
Cambridge University Press, 1994.
[5] B. M a t k o w s k i, A simple nonlinear dynamical stability problem, Bull. Amer.
Math. Soc. 76 (1970), pp. 620–625.
[6] P. G. D r a z i n, W. H. R e i d, Hydrodynamic Stability, Cambridge Monographs on
Mechanics and Applied Mathematics, Cambridge University Press, 1984.
40
V. D. Djordjević
[7] N. N. L e b e d e v, Special functions and their applications, FM, Moscow, 1963, (in
Russian)
[8] M. A b r a m o w i t z, I. A. S t e g u n, Handbook of Mathematical Functions, Dover
Publications, Inc., New York, 1972.
[9] S. L. S o b o l e v, Equations of Mathematical Physics, "Nauka", Moscow, 1966, (in
Russian).
University of Belgrade
Faculty of Mechanical Engineering
Belgrade
Serbia